 Equidecomposable subsets under a group action

For a group $G$ acting on a set $X$, subsets $A,B\subseteq X$ are $G$-equidecomposable when there are finite partitions $A={⨆}_i{A}_i$ and $B={⨆}_i{B}_i$ and elements $g_i\in G$ with $g_i{A}_i=B_i$ for every $i$.